IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
On the Exact Separation of Mixed Integer Knapsack Cuts
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
On the MIR Closure of Polyhedra
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
On the relative strength of split, triangle and quadrilateral cuts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Solving Hard Mixed-Integer Programming Problems with Xpress-MP: A MIPLIB 2003 Case Study
INFORMS Journal on Computing
Disjunctive cuts for non-convex mixed integer quadratically constrained programs
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Can pure cutting plane algorithms work?
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
On the Complexity of Selecting Disjunctions in Integer Programming
SIAM Journal on Optimization
A probabilistic analysis of the strength of the split and triangle closures
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Split Rank of Triangle and Quadrilateral Inequalities
Mathematics of Operations Research
Experiments with Two-Row Cuts from Degenerate Tableaux
INFORMS Journal on Computing
A relax-and-cut framework for gomory's mixed-integer cuts
CPAIOR'10 Proceedings of the 7th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
The chvátal-gomory closure of an ellipsoid is a polyhedron
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
On the Rank of Disjunctive Cuts
Mathematics of Operations Research
A probabilistic comparison of the strength of split, triangle, and quadrilateral cuts
Operations Research Letters
How to select a small set of diverse solutions to mixed integer programming problems
Operations Research Letters
Using symmetry to optimize over the sherali-adams relaxation
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
Approximating the Split Closure
INFORMS Journal on Computing
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The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LP-relaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the elementary split closure. This is accomplished by repeatedly solving the following separation problem: given a fractional point, say x, find a rank-1 split cut violated by x or show that none exists. Following Caprara and Letchford [17], we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program (PMILP) with a single parameter in the objective function and the right hand side. We develop an algorithmic framework to deal with the resulting PMILP by creating and maintaining a dynamically updated grid of parameter values, and use the corresponding mixed integer programs to generate rank 1 split cuts. Our approach was implemented in the COIN-OR framework using CPLEX 9.0 as a general purpose MIP solver. We report our computational results on well-known benchmark instances from MIPLIB 3.0 and several classes of structured integer and mixed integer problems. Our computational results show that rank-1 split cuts close more than 98% of the duality gap on 15 out of 41 mixed integer instances from MIPLIB 3.0. More than 75% of the duality gap can be closed on an additional 10 instances. The average gap closed over all 41 instances is 72.78%. In the pure integer case, rank-1 split cuts close more than 75% of the duality gap on 13 out of 24 instances from MIPLIB 3.0. On average, rank 1 split cuts close about 72% of the duality gap on these 24 instances. We also report results on several classes of structured problems: capacitated versions of warehouse location, single-source facility location, p-median, fixed charge network flow, multi-commodity network design with splittable and unsplittable flows, and lot sizing. The fraction of the integrality gap closed varies for these problem classes between 100 and 67%. We also gathered statistics on the average coefficient size (absolute value) of the disjunctions generated. They turn out to be surprisingly small.